Log-rank test

Builds on Statistical hypothesis test Censoring Hazard function Normal distribution Chi-squared distribution

Definition

Let two groups give right-censored lifetime data. Let $t_1 < \dots < t_m$ be the distinct event times in the pooled data. At $t_j$ , let $n_j$ be the number at risk and $d_j$ the number of events in the pooled data. Let $n_{1j}$ and $d_{1j}$ be the same numbers in group 1. Set $E_j = d_j \frac{n_{1j}}{n_j}, \qquad V_j = d_j \frac{n_{1j}}{n_j} \left(1 - \frac{n_{1j}}{n_j}\right) \frac{n_j - d_j}{n_j - 1},$ with $V_j = 0$ when $n_j = 1$ . The log-rank statistic is $Z = \frac{\sum_{j=1}^{m} (d_{1j} - E_j)}{\sqrt{\sum_{j=1}^{m} V_j}}.$ The log-rank test of the null hypothesis that the two groups have the same hazard function rejects at level $\alpha$ when $\abs{Z} > z_{1 - \alpha/2}$ . Here $z_{1 - \alpha/2}$ is the $(1 - \alpha/2)$ -quantile of the standard normal distribution.

Remarks

Under the null hypothesis and non-informative censoring, $Z$ is approximately standard normal for large samples, and $Z^2$ is approximately chi-squared with $1$ degree of freedom. Given the numbers at risk, $E_j$ and $V_j$ are the mean and variance of $d_{1j}$ under the hypergeometric distribution.

Example: group 1 has events at times $1$ and $3$ , group 2 at times $2$ and $4$ , with no censoring. The pairs $(n_j, n_{1j})$ are $(4, 2)$ , $(3, 1)$ , $(2, 1)$ , $(1, 0)$ , with $d_j = 1$ each time and $d_{1j} = 1, 0, 1, 0$ . So $\sum (d_{1j} - E_j) = 1/2 - 1/3 + 1/2 + 0 = 2/3$ and $\sum V_j = 1/4 + 2/9 + 1/4 = 13/18$ , which gives $Z \approx 0.78$ : no evidence of a difference.

Non-example: comparing the two Kaplan–Meier estimates at a single fixed time, say $t = 2$ , is a different test. It ignores the events at all other times, while the log-rank statistic combines them.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first.

Censoring concept
Chi-squared distribution concept
Construction of the reals from Cauchy sequences concept
Element definition
Hazard function concept
Integers object
Normal distribution concept
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Existential quantifier definition
Universal quantifier definition
Sample space definition
The rationals form a field statement
Cartesian product definition
Subset definition
Uniqueness quantifier definition
The rationals form an ordered field statement
Binary relation definition
Universal set definition
Power set definition
Order on the reals definition
Function definition
Complement definition
Interval definition
Preimage of a set definition
Sigma-algebra definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Independence of two events definition
Law of a random variable definition
Mutual independence of events definition
Identically distributed random variables definition
Independent sigma-algebras definition
Independence of random variables definition
Independent and identically distributed random variables definition
Statistic definition
Statistical hypothesis test concept
Log-rank test definition

Tag	`1R358D2T7J`
URI	`m2://1R358D2T7J`
Kind	definition
Writing	W0 stub
MSC	62N03, 62G10, 62P10
Defines	Log-rank test
Uses	Statistical hypothesis test, Censoring, Hazard function, Normal distribution, Chi-squared distribution

Log-rank test

Definition

Remarks

Reading path

Further reading

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